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Chapter 17: Problem 5

Solve the given inequalities. Graph each solution. $$x-3>-4$$

Short Answer

Expert verified
The solution to the inequality is \(x > -1\). Graphically, it is represented with an open circle at \(-1\) and shading to the right.

Step by step solution

01

Isolate the Variable on One Side

To solve the inequality, we need to isolate the variable \(x\). We start with the inequality: \(x - 3 > -4\). To isolate \(x\), we will add 3 to both sides of the inequality. This gives us: \(x - 3 + 3 > -4 + 3\). Simplifying this, we find \(x > -1\).
02

Graph the Solution on a Number Line

To graph the solution \(x > -1\), we draw a number line. We place an open circle at \(-1\) to indicate that \(x\) is not equal to \(-1\), but rather greater than \(-1\). We then shade the line to the right of \(-1\) to show that the solution includes all numbers greater than \(-1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequality Graphing
When dealing with inequalities, graphing the solutions helps to visually represent the set of possible values for the variable. Let's explore how graphing inequalities is done effectively.
  • First, solve the inequality so that you have a clear understanding of the solution set.
  • Identify whether the inequality is strict (using symbols like < or >) or includes equality (using \(\leq\) or \(\geq\)). For strict inequalities like \(x > -1\), use an open circle on the number line. For non-strict inequalities such as \(x \geq -1\), use a closed circle.
  • Shade the region of the number line representing the solution set. The direction of shading depends on the inequality. For \(x > -1\), shade to the right of -1, indicating all values greater than -1 are included.
By graphing, you can easily see and interpret whether values satisfy the inequality.
Isolation of Variables
Isolating the variable is a crucial step in solving inequalities. It enables you to find the solutions efficiently.
  • Start with the original inequality, such as \(x - 3 > -4\), and aim to have the variable by itself on one side.
  • Apply the same arithmetic operation on both sides of the inequality to maintain balance. For instance, adding 3 to both sides simplifies \(x - 3\) to just \(x\).
  • Be aware of operations involving multiplication or division by negative numbers, as these require flipping the inequality sign.
Once the variable is isolated, you have a simple form to identify the range of possible solutions, preparing you to graph the result.
Number Line Representation
A number line is a valuable tool for visualizing solutions to inequalities.
  • Begin by drawing a horizontal line, marking equal intervals to represent numbers.
  • Identify the critical values, like -1 in \(x > -1\), as key reference points.
  • Based on whether the solution is strict or non-strict, use open or closed circles to denote inclusivity. An open circle at -1 shows \(x\) is not equal to -1.
  • Shade the corresponding portion of the number line according to the inequality. For the example, shading towards the right signifies all values greater than -1 are part of the solution.
Using number line representation clarifies which set of numbers satisfies the inequality, aiding in understanding and communicating the solution.

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Most popular questions from this chapter

Draw a sketch of the graph of the region in which the points satisfy the given system of inequalities. $$\begin{aligned} &|y+2|<5\\\ &|x-3| \leq 2 \end{aligned}$$

Solve the given linear programming problems. An oil refinery refines types \(A\) and \(B\) of crude oil and can refine as much as 4000 barrels each week. Type A crude has 2 kg of impurities per barrel, type B has 3 kg of impurities per barrel, and the refinery can handle no more than \(9000 \mathrm{kg}\) of these impurities each week. How much of each type should be refined in order to maximize profits, if the profit is 4 dollars/barrel for type \(\mathrm{A}\) and 5 dollars/barrel for type B?

Use a calculator to display the solution of the given inequality or system of inequalities. $$\begin{aligned} &y<3-x\\\ &y>3 x-x^{3} \end{aligned}$$

Use inequalities to solve the given problems. Is \(x^{2} > x\) for all \(x ?\) Explain.

Find the indicated maximum and minimum values by the linear programming method of this section. The constraints are shown below the objective function. Minimum \(C:\) \(C=6 x+4 y\) \(y \geq 2\) \(x+y \leq 12\) \(x+2 y \geq 12\) \(2 x+y \geq 12\)
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